Dual channel reflection-type interferometers are a preferred tool used for measuring wafer shape for semiconductor processing applications. An example of such a dual channel reflection-type interferometer tool is WaferSight from KLA-Tencor. A side view diagram of a WaferSight tool 100 is shown in FIG. 1a. Wafer 102 is positioned within cavity 105. FIG. 1b shows a front view of cavity 105, showing wafer region 107 and cavity ring region 108, an annular region within the cavity but outside wafer region 107.
WaferSight uses two channels. Each measures the optical interference signal from two beams, a first reflected beam from wafer surface 110, 111 and a second reflected beam reflected from internal reference surface 120, 121, for measurement of the wafer surfaces. For Channel A interferometer (125), the phase change between wafer surface 110 and reference surface 120 is measured by using multiple phase shifted interference signals. For Channel B interferometer (130), the phase change between wafer surface 111 and reference surface 121 is measured. After the phase changes are measured by the associated interferometers, the wafer surfaces are determined as follows:ΔZ(x,y)=ΔΦ(x,y)λ(4nπ)  (1)where ΔΦ(x,y) equals the phase change, λ is the wavelength of the illuminator, n is the index of refraction of the air.
Equation (1) is not exact if the index of refraction of the air in spatially inhomogeneous. This situation occurs if the air temperature is not spatially constant, since the index of refraction of air varies with the air temperature. This relationship (i.e. between air index of refraction and air temperature) can be described empirically using the Sellmeier equation fitting curve, described at http://en.wikipedia.org/wiki/Sellmeier_formula. The exact theoretical relationship is(n−l)/(ns−1)=0.0028426*P/(T*Z)where ns is the standard index at ambient temperature and pressure,
P is pressure in Pascal,
T is temperature in degrees Kelvin,
and Z is a compressibility factor.
FIG. 2 illustrates the variation in air temperature caused by mounting a wafer with a different temperature from the tool. Wafer 200 is vertically mounted in a tool such as WaferSight, for measurement. Due to the difference between the wafer temperature and the surrounding ambient temperature of the tool region, air in top region 205 is hotter than air in bottom region 210. This induces a temperature gradient in air regions 215 and 220, due to natural convection. This temperature gradient in the air adjacent the measurement surfaces of the wafer causes a gradient in the air index of refraction, resulting in additional measured phase changes, which degrades the measurement precision.
Several prior methods have been proposed to correct the temperature-induced measurement errors:                1. U.S. Pat. No. 7,180,603, issued Feb. 20, 2007, discloses a method for reducing the additional measured optical path difference between the reference beam and the test beam by using compensating material or by mechanically mounting the interferometer to be insensitive to thermal changes.        2. U.S. Pat. No. 6,770,852, issued Aug. 3, 2004, discloses a method for controlling the local temperature of the measurement chamber in order to maintain the target critical dimension of the wafer, by using an in-situ temperature controller during the wafer etching process.        3. U.S. Pat. No. 6,924,894, issued Aug. 2, 2005, discloses a temperature compensated interferometer which maintains parity in the two beam paths (or optical fiber lengths) even with temperature perturbations.        
Each of the above proposed methods has drawbacks. All three of the methods require a separate, complicated, sub-module to control the temperature. In addition, method 2 cannot compensate the shape (e.g., line height, trench depth, sidewall angle) of the critical dimension (CD) test feature on the wafer inside the local temperature-controlled area. And method 3 requires a long settling time for stabilization of the system following heating/cooling of two fibers.